Some killers are physcology illness for killing and by killing these killers you eliminate the possibility of letting him kill again.
For example a killer may be sent for 30 years in jail and there could be either very high possibility to kill again or very small possibility of killing or not killing, so as you can see possibility of killing is two times of not killing, and by killing the killer you are sure that you will save others people lives.
The authorities are the ones who are responssible to kill the killer and not you.